Fundamental Sets of Fractal Functions

M. A. Navascués; A. K. B. Chand

doi:10.1007/s10440-007-9182-2

Fractal Frames of Functions on the Rectangle

Fractal and Fractional ◽

10.3390/fractalfract5020042 ◽

2021 ◽

Vol 5 (2) ◽

pp. 42

Author(s):

María A. Navascués ◽

Ram Mohapatra ◽

Md. Nasim Akhtar

Keyword(s):

Tensor Product ◽

Product Space ◽

Integrable Function ◽

Two Dimensional ◽

Tensor Product Space ◽

Fractal Functions

In this paper, we define fractal bases and fractal frames of L2(I×J), where I and J are real compact intervals, in order to approximate two-dimensional square-integrable maps whose domain is a rectangle, using the identification of L2(I×J) with the tensor product space L2(I)⨂L2(J). First, we recall the procedure of constructing a fractal perturbation of a continuous or integrable function. Then, we define fractal frames and bases of L2(I×J) composed of product of such fractal functions. We also obtain weaker families as Bessel, Riesz and Schauder sequences for the same space. Additionally, we study some properties of the tensor product of the fractal operators associated with the maps corresponding to each variable.

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Image Compression Using Fractal Functions

Fractal and Fractional ◽

10.3390/fractalfract5020031 ◽

2021 ◽

Vol 5 (2) ◽

pp. 31

Author(s):

Olga Svynchuk ◽

Oleg Barabash ◽

Joanna Nikodem ◽

Roman Kochan ◽

Oleksandr Laptiev

Keyword(s):

Image Compression ◽

Spatial Data ◽

Information Technologies ◽

Radio Channel ◽

Continuous Functions ◽

Interesting Property ◽

Geographic Information ◽

Fractal Image Compression ◽

Image Encoding ◽

Fractal Functions

The rapid growth of geographic information technologies in the field of processing and analysis of spatial data has led to a significant increase in the role of geographic information systems in various fields of human activity. However, solving complex problems requires the use of large amounts of spatial data, efficient storage of data on on-board recording media and their transmission via communication channels. This leads to the need to create new effective methods of compression and data transmission of remote sensing of the Earth. The possibility of using fractal functions for image processing, which were transmitted via the satellite radio channel of a spacecraft, is considered. The information obtained by such a system is presented in the form of aerospace images that need to be processed and analyzed in order to obtain information about the objects that are displayed. An algorithm for constructing image encoding–decoding using a class of continuous functions that depend on a finite set of parameters and have fractal properties is investigated. The mathematical model used in fractal image compression is called a system of iterative functions. The encoding process is time consuming because it performs a large number of transformations and mathematical calculations. However, due to this, a high degree of image compression is achieved. This class of functions has an interesting property—knowing the initial sets of numbers, we can easily calculate the value of the function, but when the values of the function are known, it is very difficult to return the initial set of values, because there are a huge number of such combinations. Therefore, in order to de-encode the image, it is necessary to know fractal codes that will help to restore the raster image.

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